TWOIPARAM ETER ISOTHERM EQUATION FOR FIBER-WATER SYSTEMS N I L S T . A N D E R S O N A N D J O S E P H L. M c C A R T H Y Department of Chemical Engineering, Unioersity of Washington, Seattle 5, Wash.
A two-parameter isotherm equation has been derived using as a basis the heat of wetting-regain relationship of Cooper and Ashpole. The equation i s identical to the de Boer-Zwicker-Bradley isotherm equation derived using polarization theory. It i s used to correlate isotherm data for 30 systems comprising synthetic fibers, natural fibers, and wood in equilibrium with water vapor. Over the range of relative humidities from 10 to 85% the standard deviation between experimental and calculated relative humidities is 1.2%, expressed in units of per cent relative humidity. Because of its simplicity and generality, the equation should be useful for the prediction, interpolation, and extrapolation of isotherm data in fiber-water systems.
relationships and isotherm equations are still and a practical standpoint. A dozen or more isotherm equations have been presented over the last 25 years. Recently, Malmquist (7 7) and Rounsley (75) have presented derivations of isotherm equations which are designed to represent isotherm data in systems containing a swellable fiber in equilibrium with water vapor. This renewed interest in isotherm equations is due in part to the fact that it is still difficult to describe fiberwater isotherm relationships over any extended relative humidity range with a simple equation. Isotherm equations presented thus far are often complex and sometimes unwieldy-e.g., the White-Eyring equation (27), which reflects the proposition that sorption occurs by a variety of overlapping steps. Among the suggested modes of sorption are : hydrate formation at low regains, multilayer sorption a t intermediate regains, and solution sorption and capillary condensation a t high regains. In a system as complex as the cellulose-water system, it is surprising that a relatively simple mathematical relationship exists between regain and heat of wetting. Such, however, is the case, as was first noted by Ashpole (2) and later by Cooper and Ashpole ( 4 ) . Their equation is IBER-WATER
F of considerable interest from both a theoretical
is the chemical potential of the adsorbate in the condensed standard state, it is possible to write pa
-
/J.:
= A R - TAS
where AH and AS are defined as the partial enthalpy and entropy, respectively, of adsorption from the liquid state. For the vapor, at ordinary pressures, the chemical potential and partial pressure are related by po =
+ R T l n PIP0
Derivation of Isotherm Equation
Equilibrium partial pressures and differential heate of adsorption are related by standard thermodynamic relationships. If pa is the chemical potential of the adsorbed species and p:
(3)
At P = Pa, = P?
and at equilibrium, Pa = Po
so that Equation 3 can be written as RT In PIP0 = A R
-
TAT
(4)
At a constant value of regain, Equation 4 can be written as (5)
For the purpose of this derivation the isosteric heat of adsorption, AB, and the differential heat of wetting, qd, are assumed identical. The differential heat of wetting dqi/dX is, from Equation 1, q d = -aqQ
where q i is the integral heat of wetting of a fiber, expressed in calories per gram of dry fiber, a t a moisture regain, X. T h e constant, qo, is the heat of wetting of the initially dry fiber. The dimensionless parameter, cy, is a n adjustable constant. We have used Equation 1 to correlate heat of wetting data for viscose rayon (79),cotton (73, 79),silk ( 6 ) ,and constituents of Eucalyptus wood (, 70) and found (7) that the average standard deviation of the heat of wetting is 0.4 cal. per gram. The range of data used corresponded to the interval 0 to 85% relative humidity. I n view of the success obtained in use of such a simple equation in correlation of heat of wetting data in these complex systems, efforts were made to derive a simple isotherm equation using, as a basis, the Cooper-Ashpole equation.
(2)
eXp(-d)
(6)
Hence Equation 5 can be written as (7)
The equation, PIP0
=
exp[ - ( a q o / R T ) exp( -ax)]
(8)
satisfies Equation 7 if cy and qo are assumed independent of temperature and if the constant of integration is taken as equal to 1. The free energy of adsorption is, from Equation 4, AF
=
RT In PIP0
=
-aqo exp( - a x )
which is identical to Equation 6. This derivation makes no allowance for A s . I n fact it appears from the derivation that A 3 is zero. This is certainly not the case, as can be seen from results presented by Morrison and Dzieciuch (73) and Wahba (79). Equation 8 is therefore rewritten as PIPa
=
exp[( - 18 PelRT) exp(&Y)]
(9)
to indicate that the constants in Equations 1 and 8 may not be identical. VOL. 2
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APRIL 1963
103
Equation 9 is identical in form to the de Boer-ZwickerBradley isotherm equation which was used by Hoover and Mellon (7) to correlate sorption data in several fiber-water systems. The de B-Z-B equation was derived using the polarization theory of adsorption which is seemingly independent of the method presented here. I t might be of theoretical interest to see if the assumptions used by Cooper and Ashpole (4) belong in the framework of the polarization theory. This point is not examined here. The constants in Equation 9 and the ones used by Hoover and Mellon (7) are related by /3 = 230.3 loglo Kz
(10)
0.1113 R T K i
(11)
@E
=
where K1 and Kz are the constants used by Hoover and Mellon. In the following section Equation 9 is used to correlate sorption data for a variety of fiber systems not studied earlier by Hoover and Mellon. Because of the lack of sorption data at various temperatures Hoover and Mellon only qualitatively indicated the temperature dependence of the isotherm parameter. This point is investigated more fully below. Hoover and Mellon did not examine the significance of the parameters appearing in the isotherm equation. This point is discussed in detail below. Presentation of Results
I t is evident from the form of Equation 9 that it is limited to the intermediate regain region. At X equal to 0, P/Po from Equation 9 is greater than zero. Also, Equation 9 predicts relative humidities less than 100%. The Cooper-Ashpole equation also deviated from experimental data in the very low and in the very high humidity regions. For these reasons the parameters p and PE were evaluated from isotherm data between 10 and 85% relative humidities.
Table 1.
System Viscose rayon Cotton linters Viscose rayon Silk Cellulose triacetate Nylon Spruce wood Sulfite pulp Cotton linters Eucalyptus wood Klason lignin Holocellulose Cellulose (from 10% NaOH) Methanol lignin Hemicellulose Cellulose (from 5% NaOH) Klinki pine
Adsorbent cotton Adsorbent cotton (desox tion) Stabilized cotton
Unstabilized cotton
104
Least squares estimates of p and PE were calculated using an IBM-709 digital computer for 30 sets of data. Equation 9 was then used to calculate values of PIP0 corresponding to specific regains. The differences between calculated and experimental values of P/Po were evaluated and are presented in Table I along with the estimates of and PE. The average standard deviation, u, in P/Po for all 30 sets of data is 0.012. Figure 1 shows the comparison between Equation 9 and experimental isotherm data for cotton, silk, and nylon. Discussion
A comparison between Equation 9 and the White-Eyring (27) and Rounsley (75)isotherm equations is shown in Figure 2. Both the White-Eyring and the Rounsley equations are threeparameter equations derived specifically to include the effects of swelling at high moisture regains. I t is evident from Figure 2 that Equation 9 is able to represent isotherm data in the high humidity region with more success compared with Rounsley’s equation. The White-Eyring equation and Equation 9 are able to represent isotherm data equally well over a large range of relative humidities. However, on the basis of ease-of-evaluation Equation 9 is to be preferred to the WhiteEyring and Rounsley equations. The adjustable parameters in both these three-parameter equations are difficult to estimate by the method of least squares, especially the White-Eyring equation. Also, it is easy to show how the parameters in Equation 9 may be estimated from independent considerations of the sorption phenomenon. There is a remarkable similarity between the constant, (- l/p), and the BET monolayer coverage, X,. For all 30 sets of data the deviation between X, and (-l/B) is less than 10% (7). If the assumption is made that (- l/p) corresponds to the number of Type 1 or strong sites in the theory of Cooper and Ashpole ( 4 ) , it can be shown that the values of p in
Parameters in Equation 9
Refer- Temp., - 8 , BE, ence a C. G./G. Cal./G.
a
25
22.10 115 18.36 126 21.13 146
0.0147 0.0164 0.0134 0.0103 0.0094 ..... 0.0112 0.0144 0.0139 0.0163 0.0168 0.0130 0.0185 0.0082
(3) 25 (3) 25
46.71 112 10.06 139
0.0159 0.0176
( 3 ) 25
22.39 18.45 19.42 19.08 55 20.70 24.6 30.74
135 128 128 113 107 131
0.0164 0.0096 0.0085 0.0082 0.0133 0.0159
26.42 29.69 31.30 31.68 31 .71 31.83 30.62 31.28 31.70
128 137 134 I35 134 132 134 129 129
0.0113 0.0052 0.0062 0.0061 0.0097 0.0103 0.0061 0.0084 0.0289
(79) 30 (79) 30 (76’1 25
’(Sj
(74)
(77) (78)
25 20 25 20
( 3 ) 25 ( 3 ) 25
(3)
(9)
(73)
18.34 31.72 18.67 22.35 41.33 49.40 18.03
i:40
24.6 0
15 27.5 40 50 15 30 40
146 134 133 134 79.6 114 124
l & E C PROCESS D E S I G N A N D DEVELOPMENT
RELATIVE HUMIDITY, 96
Figure 1. Use of isotherm equation to correlate silk, cotton, and nylon isotherm data
Table I are not unexpected. Cooper and Ashpole state that the number of Type 1 sites depends on the total regain according to the equation
where n is the number of Type 1 sites and cy is the parameter in Equation 1. At zero regain, Equation 12 becomes
where no is the initial number of Type 1 sites. I n the BET and White-Eyring theories the monolayer coverage corresponds to adsorption onto strong sites. However, X , (-1//3), so that no N ( - l / p ) . If adsorption occurs only onto Type 1 sites, then at low regains one should expect - d n / d X = 1 and hence cy = - 8. If, however, some adsorption occurs initially onto weak sites. the possibility exists that - d n / d X < 1, which implies cy < -@. This latter case is just what is observed. The value of cy for the cotton linters data reported by Wahba (79) is 21.78 ( 7 ) . From Table I the value of - p for this system is 31.72. Approximately, cy equals (3/J (- 6)for the majority of systems studied ( 7 ) . Equation 9 can be used to describe isotherm relationships roughly without any prior knowledge of equilibrium conditions in a system, if the magnitudes of 8 and @ E can be estimated. The magnitude of /3 can be estimated from our assumption regarding the correspondence between monolayer coverage and (-1//3). Also, Morrison (72) has shown how the monolayer coverage, in the cellulose-water system, corresponds to one site per accessible 6-glucopyranose residue. The magnitude of PE can be estimated from the enthalpy and entropy of adsorption since, at zero regain, AF = 1866 = AB - TAS. A good estimate of AH is the value for hydrogen bond formation in alcohol-water systems-viz.. 4500 cal. per gram mole. A good estimate of is the value for the entropy of solidification from liquid water to ice-viz., 0.292 cal. per gram O K. At 298' K. the estimate of Pe is therefore 163 cal. per gram. This value does not differ greatly from the values listed in
-
as
Table I. Of course, a better estimate of /3 and hence a more accurate predication of equilibrium conditions can be made from at least a single, experimental determination of sorption. Another illustration of the generality of this method of treating sorption data is to show that the results of Dumanski and Nekyrach (5), concerning the constancy of the average differential heat of wetting, Qd, for fibers is not unexpected. They showed, from experimental data, that Qd is 80 cal. per gram for a large number of systems. The average differential heat of wetting is given by qd
=
sr
(11x8)
Qd
(14)
d x
where X , is the regain corresponding to a value of zero for the heat of wetting. Using the Cooper-Ashpole equation q d becomes qd = qo/Xs, since e x p ( - d , ) is small compared 4(-1/6) with 1. We found that for the systems studied X , (7)-i.e., the magnitude of X , corresponds approximately to four monolayers. Also, from Equation 1: at zero regain, qo = qd(0)/cy, where qd(0) is the value of the differential heat of wetting at zero regain. Remembering that cy (3/4) (-p), qd becomes
-
qd
= qd(O)/3
The differential heat of wetting at zero regain is very nearly equal to the enthalpy of hydrogen bond formation-viz., 4500 cal. per gram mole. Hence, qd = 4500/(18) (3) or 84 cal. per gram. It appears that for systems obeying the Cooper-Ashpole equation the observations of Dumanski and Sekyrach are not unexpected. Acknowledgment
The authors thank M. Wahba for providing a large amount of isotherm data for the cotton linters-water system. They are also grateful to Rayonier Inc. for grant of a fellowship to one of us (N.T.A.). literature Cited
(1) Anderson, N. T., Ph.D. thesis, University of Washington, Seattle, Wash., 1961. (2) Ashpole, D. K., hhture 169, 37 (1952). (3) Christensen, G. N., Kelsey, K. E., Australian J . Appl. Sci. 9 , 265 (1958). (4) Cooper, D. N. E., Ashpole, D. K., J. Textile Znst. 50, T-223 (1959). (5)' Dumanski, A. V., Nekyrach, E. F., Kolloid Zhur. 15, 91 (1953). (6) Dunford, H. B., Morrison, J. L., Can. J . Chem. 33, 904 (1955). (7) Hoover, S. R., Mellon, E. F., J . Am. Chem. SOL.72, 2562 (1 o 5n\ \----/.
RELATIVE HUMIDITY, Ok
Figure 2. Comparison of equation with various isotherm equations and data for cotton linters
(8) Hutton, E. A , , Gartside, J., J. Textile Znst. 40, T-161 (1949). (9) Kelsey, K., Australian J . Appl. Sci.8, 42 (1957). (10) Kelsey, K., Christensen, G. N., Zbid., 10, 269 (1959). (11) Malmquist, L., Svenska traforskningsinst. f o r trateknik, Meddelande 98B (1958). (12) Morrison, J. L., Nature 185, 160 (1960). (13) Morrison, J. L., Dzieciuch, M. A , , Can. J . Chem. 37, 1379 (1959). (14) Ntkrsome, P. T., Sheppard, S. E., J. Phys. Chem. 36, 930 (1952). ( l i ) Rdunsley, R. R., A.Z.Ch.E. J. 7, 308 (1961). (16) Ro\ven, J. LV., Blaine, R. L., Znd. Eng. Chem. 39, 1659 (1947). (17) Speakman, J. B., Saville, A. K., J. Textzle Inst. 37, P271 (1946). (18) Stamm, A. J., IVoodruff, S . A., Znd. Eng. Chem., Anal. Ed. 13, 836 (1946). (19) Wahba, M., J . Phys. Colloid Chem. 54, 1148 (1950). (20) IVahba, M., personal communication, 1960. (21) IVhite, H. J.. Jr., Eyring, H., Textile Research J. 17, 523 (1947). RECE~VED for review January 23, 1962 ACCEPTED November 1, 1962 VOL. 2
NO. 2
APRIL
1963
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